Multiplicative Systems on Ultra-Metric Spaces

AMS Subj. Classification: MSC2010: 42C10, 43A50, 43A75

We perform analysis of certain aspects of approximation in multiplicative systems that appear as duals of ultrametric structures, e.g. in cases of local fields, totally disconnected Abelian groups satisfying the second axiom of countability or more general ultrametric spaces that do not necessarily possess a group structure. Using the fact that the unit sphere of a local field is a Vilenkin group, we introduce a new concept of differentiation in the field of p-adic numbers. Some well known convergence tests are generalized to unbounded Vilenkin groups, i.e. to the setting where the standard boundedness assumption related to the sequence of subgroups generating the underlying topology is absent. A new Fourier multiplier theorem for Hardy spaces on such locally compact groups is obtained. The strong Lq, q > 1, and weak L1 boundedness of Fourier partial sums operators in the system constructed on more general ultrametric spaces is proved.